Bernd Sturmfels

Max Planck Institute for Mathematics

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Lior Pachter

California Institute of Technology

Kathlén Kohn

KTH Royal Institute of Technology

Neil White

University of Florida

Paul Breiding

Osnabrück University

Kristian Ranestad

University of Oslo

Alicia Dickenstein

University of Buenos Aires

Günter M. Ziegler

Freie Universität Berlin

Caroline Uhler

Decision Systems (United States)

Michel Las Vergnas

Centre National de la Recherche Scientifique

Anders Björner

KTH Royal Institute of Technology

Cooperative Institutions

University of California, Berkeley

Max Planck Institute for Mathematics in the Sciences

Max Planck Institute for Mathematics

Massachusetts Institute of Technology

Max Planck Society

Technische Universität Berlin

Cornell University

ETH Zurich

Centre National de la Recherche Scientifique

University of California, Davis

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Orders and Polytropes: Matrix Algebras from Valuations

arXiv (Cornell University) (2021)

Yassine El Maazouz Marvin Anas Hahn Gabriele Nebe Mima Stanojkovski Bernd Sturmfels

We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings.

Polytrope

Convexity

Discrete valuation

Tropical Geometry

Matrix (chemical analysis)

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Hypergeometric Series

Algorithms and computation in mathematics (2000)

Mutsumi Saito Bernd Sturmfels Nobuki Takayama

Basic hypergeometric series

Bilateral hypergeometric series

Appell series

Hypergeometric identity

Lauricella hypergeometric series

Barnes integral

Hypergeometric distribution

10.1007/978-3-662-04112-3_3

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3264 Conics in a Second

arXiv (Cornell University) (2019)

Paul Breiding Bernd Sturmfels Sascha Timme

Enumerative algebraic geometry counts the solutions to certain geometric constraints. Numerical algebraic geometry determines these solutions for any given instance. This article illustrates how these two fields complement each other. Our focus lies on the 3264 conics that are tangent to five given conics in the plane. We present a web interface for computing them. It uses the software HomotopyContinuation.jl, which makes this process fast and reliable. We discuss an instance where all 3264 solutions are real.

Conic section

Complement

10.48550/arxiv.1902.05518

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General Models for Rational Cameras and the Case of Two-Slit Projections

Matthew Trager Bernd Sturmfels John Canny Martial Hebert Jean Ponce

The rational camera model recently introduced in [18] provides a general methodology for studying abstract nonlinear imaging systems and their multi-view geometry. This paper builds on this framework to study physical realizations of rational cameras. More precisely, we give an explicit account of the mapping between between physical visual rays and image points (missing in the original description), which allows us to give simple analytical expressions for direct and inverse projections. We also consider primitive camera models, that are orbits under the action of various projective transformations, and lead to a general notion of intrinsic parameters. The methodology is general, but it is illustrated concretely by an in-depth study of two-slit cameras, that we model using pairs of linear projections. This simple analytical form allows us to describe models for the corresponding primitive cameras, to introduce intrinsic parameters with a clear geometric meaning, and to define an epipolar tensor characterizing two-view correspondences. In turn, this leads to new algorithms for structure from motion and self-calibration.

Epipolar geometry

10.1109/cvpr.2017.270

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The Graph of Monomial Ideals

arXiv (Cornell University) (2002)

Klaus Altmann Bernd Sturmfels

There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring. The definition involves Gr\obner bases or the action of an algebraic torus. We present algorithms for computing the (affine schemes representing) edges in this graph. We study the induced subgraphs on multigraded Hilbert schemes and on square-free monomial ideals. In the latter case, the edges correspond to generalized bistellar flips.

Square-free integer

Monomial ideal

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Tensoren und ihre Zerlegungen

Mateusz Michałek Bernd Sturmfels

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Real rank two geometry

Journal of Algebra (2017)

Anna Seigal Bernd Sturmfels

Rank (graph theory)

Algebraic variety

10.1016/j.jalgebra.2017.04.014

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Cellular resolutions of monomial modules

Journal für die reine und angewandte Mathematik (Crelles Journal) (1998)

Dave Bayer Bernd Sturmfels

We construct a canonical free resolution for arbitrary monomial modules and lattice ideals.This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals [BPS], [PS].

10.1515/crll.1998.083

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Citations (209)

Distortion Varieties

arXiv (Cornell University) (2016)

Joe Kileel Zuzana Kúkelová Tomáš Pajdla Bernd Sturmfels

The distortion varieties of a given projective variety are parametrized by duplicating coordinates and multiplying them with monomials. We study their degrees and defining equations. Exact formulas are obtained for the case of one-parameter distortions. These are based on Chow polytopes and Gröbner bases. Multi-parameter distortions are studied using tropical geometry. The motivation for distortion varieties comes from multi-view geometry in computer vision. Our theory furnishes a new framework for formulating and solving minimal problems for camera models with image distortion.

Distortion (music)

10.48550/arxiv.1610.01860

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Condition Numbers

Oberwolfach Seminars (2024)

Paul Breiding Kathlén Kohn Bernd Sturmfels

Abstract The concept of a condition number has its origin in numerical analysis. It measures how much the output value of a function we wish to evaluate can change for a small change in the input argument. In this chapter, we discuss condition numbers in the context of metric algebraic geometry. In the first section, we offer an introduction to the relevant notions for assessing errors in numerical computations.

10.1007/978-3-031-51462-3_9

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